A Bayesian Perspective on Structured Mixtures of IRT Models
-
Upload
boris-walter -
Category
Documents
-
view
28 -
download
3
description
Transcript of A Bayesian Perspective on Structured Mixtures of IRT Models
![Page 1: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/1.jpg)
CILVR 2006 Slide 1May 18, 2006
A Bayesian Perspective on Structured Mixtures of IRT Models
Robert Mislevy, Roy Levy, Marc Kroopnick, and Daisy Wise
University of Maryland
Presented at the Conference “Mixture Models in Latent Variable Research,” May 18-19, 2006, Center for Integrated Latent Variable Research, University of Maryland
![Page 2: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/2.jpg)
CILVR 2006 Slide 2May 18, 2006
Probability is not really about numbers;
it is about the structure of reasoning. Glen Shafer, quoted in Pearl, 1988, p. 77
![Page 3: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/3.jpg)
CILVR 2006 Slide 3May 18, 2006
Where we are going
The structure of assessment arguments Probability-based reasoning in assessment Increasingly complex psychological narratives
entail…» Extended view of “data”» More encompassing probability models,
from Classical test theory to mixtures of structured Item response theory (IRT) models
![Page 4: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/4.jpg)
CILVR 2006 Slide 4May 18, 2006
The structure of assessment arguments
A construct-centered approach would begin by asking what complex of knowledge, skills, or other attribute should be assessed, …
Next, what behaviors or performances should reveal those constructs, and
what tasks or situations should elicit those behaviors?
Messick, 1992, p. 17
![Page 5: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/5.jpg)
CILVR 2006 Slide 5May 18, 2006
An Example: Mental Rotation Tasks
Stimulus
![Page 6: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/6.jpg)
CILVR 2006 Slide 6May 18, 2006
An Example: Mental Rotation Tasks
Target
![Page 7: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/7.jpg)
CILVR 2006 Slide 7May 18, 2006
Total Scores• Counts of multiple observations which all signify
proficiency--More is better. • Everyone takes same tasks. • Comparisons/decisions based on total scores X. • Nature of tasks outside the model.• No distinction between observation (X) and target of
inference (proficiency at mental rotation);• No probability-based model for characterizing evidence.• No notion of “measurement error” (except when your
score is lower than you think it ought to be)
![Page 8: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/8.jpg)
CILVR 2006 Slide 8May 18, 2006
A properly-structured statistical model overlays a substantive model for the situation with a model for our knowledge of the situation,
so that we may characterize and communicate what we come to believe—as to both content and conviction—and why we believe it—as to our assumptions, our conjectures, our evidence, and the structure of our reasoning.
Mislevy & Gitomer, 1996
Enter probability-based reasoning
![Page 9: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/9.jpg)
CILVR 2006 Slide 9May 18, 2006
Defining variables
A frame of discernment is all the distinctions one can make within a particular model (Shafer, 1976). » “To discern” means “to become aware of” and “to
make distinctions among.” In assessment, the variables relate to the claims
we would like to make about students and the observations we need to make.
All are framed and understood in terms appropriate to the purpose, the context, and psychological perspective that ground the application.
![Page 10: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/10.jpg)
CILVR 2006 Slide 10May 18, 2006
Conditional independence In assessment, the statistical concept of conditional
independence formalizes the working assumption that if the values of the student model variables were known, there would have no further information in the details.
We use a model at a given grainsize or with certain kinds of variables not because we think that is somehow ”true”, but rather because it adequately expresses patterns in the data in light of our perspective on knowledge/skill and the purpose of the assessment.
![Page 11: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/11.jpg)
CILVR 2006 Slide 11May 18, 2006
Classical Test Theory (CTT) Still total scores, but with the idea of
replication—multiple “parallel” tests Xj that may differ, but are all “noisy” versions of the same “true score” :
Xij = i + eij ,
where eij ~ N(0, e2).
Details of cognition, observation task by task, and content of tasks lie outside the probability model.
![Page 12: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/12.jpg)
CILVR 2006 Slide 12May 18, 2006
Classical Test Theory (CTT)
Directed graph representation of Bayesian probability model for multiple parallel tests.
Note direction of conditional probability in model; contrast w. inference about once Xs are observed.
X 1 X 2 X 3
p( )
p( X 1 | )
p( X 2 | )
p( X 3 | )
![Page 13: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/13.jpg)
CILVR 2006 Slide 13May 18, 2006
Classical Test Theory (CTT)
2 2 *
* 2 2 2 2
, , ,
| , | , .
e
ij i e i ei j
p
p x p p p p
x
Posterior inference, via Bayes theorem:
2 2
2 2 2 2
, , , ,
| , | , .
e
ij i e i ei j
p
p x p p p p
x
The full probability model:
Note what’s there, conditional independence, relationships, what’s not there.
![Page 14: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/14.jpg)
CILVR 2006 Slide 14May 18, 2006
Classical Test Theory (CTT) For Student i, posterior inference is probability distribution for i; that
is,» Expected value for test score, along with standard deviation of distribution for
i .» People can be seen as differing only as to propensity to make correct
responses. Inference bound to particular test form. Also posterior distribution for mean & variance of s, error variance,
and reliability, or
2 2 2 .e
![Page 15: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/15.jpg)
CILVR 2006 Slide 15May 18, 2006
Item Response Theory (IRT)
Modeling now at level of items» Adaptive testing, matrix sampling, test assembly
Multiple non-parallel item responses Xij that may differ, but all depend on i:
where j is the possibly vector-valued parameter of Item j.
Conditional independence of item responses given and parameter(s) for each item.
1 1Pr ,..., , ,..., Pr , ,i iJ i J ij i jj
x x x
![Page 16: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/16.jpg)
CILVR 2006 Slide 16May 18, 2006
Item Response Theory (IRT)
Same item ordering for all people. Content of tasks still outside the probability model.
easier harder
The Rasch IRT model for 0/1 items:
Prob(Xij=1|i,j) = (i - j), where(x) = exp(x)/[1+exp(x)].
Item 1 Item 4 Item 5 Item 3 Item 6 Item 2
Person A Person B Person D
less able more able
![Page 17: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/17.jpg)
CILVR 2006 Slide 17May 18, 2006
A full Bayesian model for IRT
, , , , | , | |ij i j i ji j
p X p X p p p p The measurement model:Item responses conditionally independentgiven person & item parameters
Distributions for person parameters
Distributions for item parameters
Distribution for parameter(s) of distributions for item parameters
Distribution for parameter(s) of distributions for person parameters
![Page 18: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/18.jpg)
CILVR 2006 Slide 18May 18, 2006
A full Bayesian model for IRT
*
*
, , ,
| , | |ij i j i ji j
p
p x p p p p
x
Posterior inference:
•For people, posterior distributions for s, or propensity to make correct responses.•How/why outside model. •For items, posterior distributions for s. •Some harder, some easier; How/why outside model.•Presumed to be the same for all people.
![Page 19: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/19.jpg)
CILVR 2006 Slide 19May 18, 2006
Summary test scores … have often been though of as “signs” indicating the presence of underlying, latent traits.
An alternative interpretation of test scores as samples of cognitive processes and contents, and of correlations as indicating the similarity or overlap of this sampling, is equally justifiable and could be theoretically more useful.
The evidence from cognitive psychology suggests that test performances are comprised of complex assemblies of component information-processing actions that are adapted to task requirements during performance.
(Snow & Lohman, 1989, p. 317)
The “cognitive revolution”
![Page 20: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/20.jpg)
CILVR 2006 Slide 20May 18, 2006
Research on mental rotation tasks
Roger Shepard’s studies in the early 70’s shows difficulty of mental rotation tasks depends mainly on how much it is rotated.
Shepard, R. N. & Meltzer, J (1971) Mental rotation of three-dimensional objects. Science, 171, 701-703.
Cooper, L. A., & Shepard, R. N. (1973). Chronometric studies of the rotation of mental images. In W. G. Chase (Ed.), Visual Information Processing (pp. 75–176). New York: Academic Press.
![Page 21: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/21.jpg)
CILVR 2006 Slide 21May 18, 2006
A structured IRT model: The LLTM
The linear logistic test model (Fischer, 1973) Rasch model but item parameters conditional on item
features qj:
where k is a contribution to difficulty from feature k.• Now difficulty modeled as function of task features, as
correlated with demands for aspects of knowledge or processing. Conditional independence of item parameters given features: They “explain” item difficulty.
• Task features bring psychological theory into model.• For mental rotation, can use degree of rotation for qj .
,j jk k jk
q q
![Page 22: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/22.jpg)
CILVR 2006 Slide 22May 18, 2006
Content of tasks inside the probability model.
easier harder
Item 1 Item 4 Item 5 Item 3 Item 6 Item 2
Person A Person B Person D
less able more able
Less rotation More rotation
A structured IRT model: The LLTM
![Page 23: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/23.jpg)
CILVR 2006 Slide 23May 18, 2006
A full Bayesian model for LLTM
, , , , ,
| , | | ,ij i j i j ji j
p X q
p X p p q p p
The measurement model:Item responses conditionally independentgiven person & item parameters
Distributions for item parametersconditional on item features
Note on extensions:•Rating scale, count, vector-valued observations.•Multivariate student models for multiple ways people differ; conditional independence given vector . •Different item features relevant to different components of
![Page 24: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/24.jpg)
CILVR 2006 Slide 24May 18, 2006
A full Bayesian model for LLTM
*
*
, , , ,
| , | | , .ij i j i j ji j
p q
p x p p q p p
x
Posterior inference:
•For people, posterior distributions for s, or propensity to make correct responses.•How/why interpreted through psychological model. •For items, posterior distributions for s. •Still resumed to be the same for all people.•Some harder, some easier; How/why inside model.•Hypothesized patterns can be checked statistically.
![Page 25: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/25.jpg)
CILVR 2006 Slide 25May 18, 2006
Structured mixtures of IRT models
What makes items hard may depend on solution strategy. » John French (1965) on mental rotation.» Siegler re balance beam--development. » Gentner & Gentner on electrical circuits.» Tatsuoka on mixed number subtraction
Theory says what the relationship ought to be; the trick is putting it into a probability model.
![Page 26: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/26.jpg)
CILVR 2006 Slide 26May 18, 2006
Structured mixtures of IRT models
For mental rotation items: Difficulty depends on angle of rotation if mental
rotation strategy. Difficulty depends on acuteness of angle if analytic
strategy. Can make inferences about group membership using
items that are relatively hard under one strategy, relatively easy under the other.
Mislevy, R.J., Wingersky, M.S., Irvine, S.H., & Dann, P.L. (1991).
Resolving mixtures of strategies in spatial visualization tasks.
British Journal of Mathematical and Statistical Psychology, 44,
265-288.
![Page 27: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/27.jpg)
CILVR 2006 Slide 27May 18, 2006
easier harder
Item 1 Item 4 Item 5 Item 3 Item 6 Item 2
Person A Person B Person D
less able rotatemore
able
Less rotation More rotation
A structured IRT mixture
easier harder
Item 4 Item 2 Item 6 Item 1 Item 5 Item 3
Person B Person A Person D
less able analyticmore
able
More acute Less acute
OR
![Page 28: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/28.jpg)
CILVR 2006 Slide 28May 18, 2006
Structured mixtures of IRT models
Groups of people distinguished by the way they solve tasks.
ik =1 if Person i is in Group k, 0 if not. People differ as to knowledge, skills, proficiencies
within group, expressed by ik’s. Items differ as to knowledge, skills, demands within
group, expressed by qjk’s. Thus, LLTM models within groups. Conditional independence of responses given
person, item, and group parameters.
![Page 29: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/29.jpg)
CILVR 2006 Slide 29May 18, 2006
Structured mixtures of IRT models
Consider M strategies; each person applies one of them to all items, and item difficulty under strategy m depends on features of the task that are relevant under this strategy in accordance with an LLTM structure.
The difficulty of item j under strategy m is The probability of a correct response is
jm jmk mkk
b q
Pr 1 , , , .i
ij i i j im jmk mkkm
X q q
![Page 30: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/30.jpg)
CILVR 2006 Slide 30May 18, 2006
Structured mixtures of IRT models
1
, , , , , ,
| , , | | ,ik
ij ik jk i ik k i jk jk k k ki j k
p X q
p X p p p q p p
Item responses conditionally independentgiven person’s group and person & item parameters relevant to that group.
Distributions for item parametersconditional on item features and feature effects relevant to each group
![Page 31: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/31.jpg)
CILVR 2006 Slide 31May 18, 2006
Structured mixtures of IRT models
*
*
1
, , , , , ,
| , , | | ,ik
ij ik jk i ik k i jk jk k k ki j k
p q
p x p p p q p p
x
Posterior inference:
•For people, posterior probs for s, or group memberships.•Posterior distributions for s within groups.•How/why interpreted through psychological model. •For items, posterior distributions for s for each group. •Items differentially difficult for different people, based on theories about what makes items hard under different strategies.
![Page 32: A Bayesian Perspective on Structured Mixtures of IRT Models](https://reader036.fdocuments.net/reader036/viewer/2022062718/56812f64550346895d94ee2e/html5/thumbnails/32.jpg)
CILVR 2006 Slide 32May 18, 2006
Conclusion
Evolution of psychometric models joint with psychology.
Bayesian probability framework well suited for building models that correspond to “narratives”
Can’t just “throw data over the wall” like was done with CTT;
Need to build coordinated observational model and probability model, from psychological foundation.